Optimal. Leaf size=48 \[ -\frac {a^4}{x}+4 a^3 b x+2 a^2 b^2 x^3+\frac {4}{5} a b^3 x^5+\frac {b^4 x^7}{7} \]
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Rubi [A]
time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {28, 276}
\begin {gather*} -\frac {a^4}{x}+4 a^3 b x+2 a^2 b^2 x^3+\frac {4}{5} a b^3 x^5+\frac {b^4 x^7}{7} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 276
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^2} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^4}{x^2} \, dx}{b^4}\\ &=\frac {\int \left (4 a^3 b^5+\frac {a^4 b^4}{x^2}+6 a^2 b^6 x^2+4 a b^7 x^4+b^8 x^6\right ) \, dx}{b^4}\\ &=-\frac {a^4}{x}+4 a^3 b x+2 a^2 b^2 x^3+\frac {4}{5} a b^3 x^5+\frac {b^4 x^7}{7}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 48, normalized size = 1.00 \begin {gather*} -\frac {a^4}{x}+4 a^3 b x+2 a^2 b^2 x^3+\frac {4}{5} a b^3 x^5+\frac {b^4 x^7}{7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 45, normalized size = 0.94
method | result | size |
default | \(-\frac {a^{4}}{x}+4 a^{3} b x +2 a^{2} b^{2} x^{3}+\frac {4 a \,b^{3} x^{5}}{5}+\frac {b^{4} x^{7}}{7}\) | \(45\) |
risch | \(-\frac {a^{4}}{x}+4 a^{3} b x +2 a^{2} b^{2} x^{3}+\frac {4 a \,b^{3} x^{5}}{5}+\frac {b^{4} x^{7}}{7}\) | \(45\) |
norman | \(\frac {\frac {1}{7} b^{4} x^{8}+\frac {4}{5} a \,b^{3} x^{6}+2 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}-a^{4}}{x}\) | \(48\) |
gosper | \(-\frac {-5 b^{4} x^{8}-28 a \,b^{3} x^{6}-70 a^{2} b^{2} x^{4}-140 a^{3} b \,x^{2}+35 a^{4}}{35 x}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 44, normalized size = 0.92 \begin {gather*} \frac {1}{7} \, b^{4} x^{7} + \frac {4}{5} \, a b^{3} x^{5} + 2 \, a^{2} b^{2} x^{3} + 4 \, a^{3} b x - \frac {a^{4}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 48, normalized size = 1.00 \begin {gather*} \frac {5 \, b^{4} x^{8} + 28 \, a b^{3} x^{6} + 70 \, a^{2} b^{2} x^{4} + 140 \, a^{3} b x^{2} - 35 \, a^{4}}{35 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 44, normalized size = 0.92 \begin {gather*} - \frac {a^{4}}{x} + 4 a^{3} b x + 2 a^{2} b^{2} x^{3} + \frac {4 a b^{3} x^{5}}{5} + \frac {b^{4} x^{7}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 7.39, size = 44, normalized size = 0.92 \begin {gather*} \frac {1}{7} \, b^{4} x^{7} + \frac {4}{5} \, a b^{3} x^{5} + 2 \, a^{2} b^{2} x^{3} + 4 \, a^{3} b x - \frac {a^{4}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 44, normalized size = 0.92 \begin {gather*} \frac {b^4\,x^7}{7}-\frac {a^4}{x}+\frac {4\,a\,b^3\,x^5}{5}+2\,a^2\,b^2\,x^3+4\,a^3\,b\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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